Finite Group Databasebeta

← #27715

Subgroups — #27715

Order 256.

367 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2502C211yesmaximal, characteristic
2#2442C211yesmaximal, characteristic
3#2099C211yesmaximal, characteristic, complemented
4#2099C211yesmaximal, characteristic, complemented
5#2074C211yesmaximal, characteristic, complemented
6#2026C211yesmaximal, characteristic, complemented
7#1992C211yesmaximal, characteristic, complemented
8#1935C211yesmaximal, characteristic, complemented
9#1935C211yesmaximal, characteristic, complemented
10#1892C211yesmaximal, characteristic
11#1872C211yesmaximal, characteristic
12#1836C211yesmaximal, characteristic, complemented
13#1824C211yesmaximal, characteristic
14#1752C211yesmaximal, characteristic
15C2×#340C211yesmaximal, characteristic
16C23×C8C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C2×#103C2211yescharacteristic
22C2×#103C2211yescharacteristic
23C2×#103C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×C22⋊C8C2211yescharacteristic
26C2×#97C2211yescharacteristic
27C2×#97C2211yescharacteristic
28#400—21no—
29#398—21no—
30#398—21no—
31#398—21no—
32#391C2211yescharacteristic
33#391C2211yescharacteristic
34#389C2211yescharacteristic
35#389C2211yescharacteristic
36#385—21no—
37#385—21no—
38#385—21no—
39#384C2211yescharacteristic
40#384C2211yescharacteristic
41#382C2211yescharacteristic
42#380—21no—
43#380C2211yescharacteristic
44C4×C22⋊C4C2211yescharacteristic
45#340C2211yescharacteristic
46#340C2211yescharacteristic
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C23×C4—21no—
61C23×C4—21no—
62C23×C4C2311yescharacteristic
63C22×C8—22no—
64C22×C8—22no—
65C22×C8—41no—
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70#125—22no—
71#125—41no—
72C22⋊Q8—22no—
73C22⋊Q8—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4D411yescharacteristic
99C2×C22⋊C4D411yescharacteristic
100C2×C42C2311yescharacteristic
101C2×C42C2311yescharacteristic
102#103—22no—
103#103—22no—
104#103—41no—
105#103—41no—
106C22⋊C8—22no—
107#97—22no—
108#97—41no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95D411yescharacteristic
119#95D411yescharacteristic
120C24C2×D411yescharacteristic
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—81no—
143C22×C4—81no—
144C22×C4—81no—
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4#4112yes—
151C22×C4#4112yes—
152C22×C4C2411yescharacteristic
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—44no—
165C2×C8#4112yes—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—81no—
185C4⋊C4—81no—
186C4⋊C4—81no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4—81no—
190C4⋊C4#4112yes—
191C22⋊C4—22no—
192C22⋊C4—22no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—21no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23#12412yes—
204C23#13712yes—
205C23#13712yes—
206C23C22×D411yescharacteristic
207Q8—41no—
208Q8—41no—
209Q8—41no—
210Q8—41no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—84no—
254C2×C4—84no—
255C2×C4#12412yes—
256C2×C4C2×#4111yescharacteristic
257C2×C4C2×#4111yescharacteristic
258C2×C4C2×#4111yescharacteristic
259C2×C4C2×#4111yescharacteristic
260C2×C4H(5,𝔽2)11yescharacteristic
261C2×C42−1+411yescharacteristic
262C8—22no—
263C8—41no—
264C8—41no—
265C8—41no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22#49114yes—
273C22C2×#12411yescharacteristic
274C22#54611yescharacteristic
275C22#54711yescharacteristic
276C22C2×#13711yescharacteristic
277C22C2×#13711yescharacteristic
278C22#57611yescharacteristic
279C22#57711yescharacteristic
280C4—22no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—42no—
285C4—41no—
286C4—81no—
287C4—81no—
288C4—81no—
289C4—81no—
290C4—81no—
291C4—81no—
292C4—81no—
293C4#52911yescharacteristic
294C4#53911yescharacteristic
295C4#55111yescharacteristic
296C4#55211yescharacteristic
297C2—24no—
298C2#248811yescharacteristic
299C2C2×#49111yescharacteristic
300C2#313611yescharacteristic
301C2#331411yescharacteristic
302C2#331611yescharacteristic
303C2#334411yescharacteristic
304C2#334611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #2099C2semidirect
4. #2099C2semidirect
5. #2074C2semidirect
6. #2026C2semidirect
7. #1992C2semidirect
8. #1935C2semidirect
9. #1935C2semidirect
12. #1836C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#25022
#24422
#20992
#20742
#20262
#19922
#19352
#18922
#18722
#18362
#18242
#17522
C2×#3402